Current methods of analyzing astigmatism are confined to calculation of the vector of change surgically induced in attaining the post-operative result from the pre-operative state.
This ably allows determination of total induced astigmatism and the direction of the vector force acting in the eye. It also enables calculation of the mean total surgical astigmatism induced when a series of operations are compared and analyzed. However, the axes of surgical induced astigmatism (SIA) generally varies considerably within the 180.degree. arc of range. This makes it extremely difficult to make meaningful comparisons of astigmatic change for a series, as one cannot obtain an average directional change of vectors, as vectors in opposing or partly opposing directions cancel each other out in varying amounts.
One practice carried out by some surgeons is to resort to the sole option of tabulating each patient's results individually, leaving it to the reader to estimate any trend. Some surgeons attempt to provide an overview of results, but lack the means to deduce a trend in induced astigmatism vectors as a group, because they have variable orientation.
Taking a mean of the angles has no validity in determining the trend for axes, nor does it address the change in axes from their pre-operative to post-operative astigmatic status. It does not assess the success or desirability of the achieved result; furthermore, it does not indicate the extent to which the surgical aim was achieved. An attempt has been made to address the complexities of correcting the magnitude for the degrees of axis change by introducing the approximation that this component varies as the cosine of the difference between the attempted and the observed (achieved) axes. This corrected value of magnitude was substituted as the amount of surgically induced astigmatism measured on a cylinder 90.degree. to the axis of the incisions, the so-called "proper" axis. It has been proposed to program so called Naylor's equations into a computer program that requires slight modifications to resolve the ambiguity and essentially reproduce the Naylor table.
The formula for calculation of SIA is derived from the resultant of two plano-cylindrical lenses with axes at different angles; this was subsequently employed by some surgeons using graphical methods confirming the magnitude and axis of the astigmatic change. Jaffe and Clayman employ rectangular and polar co-ordinates to determine, by vector analysis, the formula for calculating SIA and its axis with the known values for pre- and post-operative corneal astigmatism. Analogous formulae were derived by Hall based on Martin and Welford's derivation of Euler's theorem of curved surfaces (investigated by Airy in 1827).
Euler's theorem, which states "that the sum of the curvatures of any two perpendicular sections of a cylindrical or toric surface has a constant value", provides the link between Jaffe's and Naeser's methods of vector analysis. Naeser's method calculates the polar values of astigmatism, arising when the axis of astigmatism does not lie on 90.degree. or 180.degree. meridia; its use lies primarily in interpreting results of surgery which induces polar (with-the-rule and against-the-rule) changes, such as cataract and implant surgery (with or without transverse astigmatic keratotomy).
Astigmatism is a unique refractive error that causes reduced visual acuity and produces symptoms such as glare, monocular diplopia, asthenopia and distortion. For some years now, astigmatism control and correction has been of great concern to refractive, cataract and corneal surgeons. Reduction or elimination of astigmatism, as a single or combined procedure, is only possible if one possesses an understanding of astigmatic change, in its component parts of magnitude and axis. Current analytical techniques do not allow us to compare magnitudes and axes separately for a series of paired groups of procedures or for a single procedure, yet it is only in this way that we are able to perfect techniques of astigmatism surgery. We need to be able to determine the preferable technique to employ; we also need to be able to determine whether any failure to achieve surgical goals is attributable to an individual patient factor or to machine or technique error. Modern laser technologies have empowered us with the ability to modify our procedures with degrees of sophistication not previously possible; this in turn requires analysis systems which will allow us to accurately quantify and scientifically assess the results.